1
$\begingroup$

Let $f$ be a holomorphic function on unit disk $\mathbb{D}$ such that $f(0)=0$ and there exists constant $A>0$ such that $\operatorname{Re}(f(z)) \leq A$, for all $z \in \mathbb{D}.$ Could anyone advise me how to prove $|f(z)| \leq \dfrac{2A|z|}{1-|z|}, \forall z\in \mathbb{D \ ?}$ I tried using the theorem of Schwarz pick to no avail. Hints will suffice, thank you very much.

$\endgroup$
  • $\begingroup$ I see your point. $\endgroup$ – AnyAD Oct 6 '14 at 10:46
2
$\begingroup$

More generally: suppose $f:\mathbb D\to \Omega$ is a holomorphic map into a simply connected domain $\Omega$, and let $\phi:\mathbb D\to\Omega$ be a bijective holomorphic map such that $\phi(0)=f(0)$. By the Schwarz lemma applied to $\phi^{-1}\circ f$ we have $(\phi^{-1}\circ f)( \mathbb D_r)\subset \mathbb D_r$ for $0<r<1$. Rewrite this as $f(\mathbb D_r)\subset \phi(\mathbb D_r)$, and it begins to look promising.

To find $\phi$ for your particular domain, think of fractional linear transformations.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.